Degenerate Cahn-Hilliard and incompressible limit of a Keller-Segel model

TL;DR

This paper investigates the convergence of a generalized Keller-Segel chemotaxis model to a degenerate Cahn-Hilliard system, providing insights into modeling living tissues with compressible and incompressible properties.

Contribution

It establishes the equivalence between Keller-Segel and a relaxed degenerate Cahn-Hilliard system and analyzes the incompressible limit of the model.

Findings

Proves convergence of Keller-Segel to Cahn-Hilliard system.

Analyzes the incompressible limit of the model.

Provides a mathematical framework for tissue modeling.

Abstract

The Keller-Segel model is a well-known system representing chemotaxis in living organisms. We study the convergence of a generalized nonlinear variant of the Keller-Segel to the degenerate Cahn-Hilliard system. This analysis is made possible from the observation that the Keller-Segel system is equivalent to a relaxed version of the Cahn-Hilliard system. Furthermore, this latter equivalent system has an interesting application in the modelling of living tissues. Indeed, compressible and incompressible porous medium type equations are widely used to describe the mechanical properties of living tissues. The relaxed degenerate Cahn-Hilliard system, can be viewed as a compressible living tissue model for which the movement is driven by Darcy's law and takes into account the effects of the viscosity as well as surface tension at the surface of the tissue. We study the convergence of the Keller-Segel system to the Cahn-Hilliard equation and some of the analytical properties of the model such as the incompressible limit of our model. Our analysis relies on a priori estimates, compactness properties, and on the equivalence between the Keller-Segel system and the relaxed degenerate Cahn-Hilliard system.